puutcharacteristic

Puutcharacteristic is a theoretical invariant used in graph theory to quantify the branching pattern of tree-like structures. The concept is defined for rooted trees and is intended to provide a compact representation of a tree’s depth distribution.

Definition: For a rooted tree T with root r, the puutcharacteristic P(T) is a depth-profile vector (p0, p1, ..., ph), where pk is the number of nodes at distance k from r; h is the height of T. This vector is invariant under rooted-tree isomorphisms. For unrooted trees, the profile depends on the choice of root; in practice one may consider the multiset of profiles over all possible roots.

Variants and scalar measures: A weighted version P_w(T) uses a nonnegative sequence {w_k} to weight levels, producing a scalar value sum_k w_k pk, called the puutvalue. Normalized forms divide by the total number of nodes to compare trees of different sizes.

Computation and examples: P(T) is computed by a breadth-first traversal. A path with five vertices rooted at an end has profile (1,1,1,1,1). A star with one center and n leaves has profile (1,n).

Applications and relation: The puutcharacteristic serves as a simple descriptor for tree similarity, clustering, and phylogenetic analyses where depth distribution matters. It relates to tree height and branching factors, and can complement other invariants like degree sequence and path-length distributions.

Origin and usage: The term is a coinage used in theoretical discussions and did not enter formal standard terminology. The word blends "puu" (tree) with "characteristic" to emphasize the tree-structure focus of the invariant.